$\frac{\cos 2\alpha }{{\cos }^4\alpha -{\sin }^4\alpha }-\frac{{\cos }^4\alpha +{\sin }^4\alpha }{2-{\sin }^{2}2\alpha }=$

$\begin{array}{rcl} & & \frac{\cos 2\mathrm{\alpha }}{{\cos }^4\mathrm{\alpha }-{\sin }^4\mathrm{\alpha }}-\frac{{\cos }^4\mathrm{\alpha }+{\sin }^4\mathrm{\alpha }}{2-{\sin }^{2}2\mathrm{\alpha }}\\ & =& \frac{\cos 2\mathrm{\alpha }}{\left({\cos }^2\mathrm{\alpha }-{\sin }^2\mathrm{\alpha }\right) \left({\cos }^2\mathrm{\alpha }+{\sin }^2\mathrm{\alpha }\right)}\\ & & -\frac{{\left({\cos }^2\mathrm{\alpha }+{\sin }^2\mathrm{\alpha }\right)}^2-2{\sin }^2\mathrm{\alpha } {\cos }^2\mathrm{\alpha }}{2-4 {\sin }^2\mathrm{\alpha } {\cos }^2\mathrm{\alpha }}\\ & =& \frac{\cos 2\mathrm{\alpha }}{\cos 2\mathrm{\alpha }}-\frac{1-2{\sin }^2\mathrm{\alpha } {\cos }^2\mathrm{\alpha }}{2\left(1-2{\sin }^2\mathrm{\alpha } {\cos }^2\mathrm{\alpha }\right)}\\ & =& 1-\frac{1}{2}=\frac{1}{2}\end{array}$